CN107609573A - High spectrum image time varying characteristic extracting method based on low-rank decomposition and empty spectrum constraint - Google Patents

High spectrum image time varying characteristic extracting method based on low-rank decomposition and empty spectrum constraint Download PDF

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CN107609573A
CN107609573A CN201710706837.1A CN201710706837A CN107609573A CN 107609573 A CN107609573 A CN 107609573A CN 201710706837 A CN201710706837 A CN 201710706837A CN 107609573 A CN107609573 A CN 107609573A
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陈昭
卢婷
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Donghua University
National Dong Hwa University
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Abstract

The present invention relates to a kind of high spectrum image time varying characteristic extracting method based on low-rank decomposition and empty spectrum constraint, mainly include the following steps that：Calculate the differential image in two groups of high-spectrum remote sensings of the same place collection of different time；The low-rank matrix decomposition model with empty spectrum constraint is established using the inherent data structure characteristics of differential image；Each component of the model is solved by way of alternating iteration, and then extracts time domain variation characteristic.The present invention makes full use of the immanent structure of data, it is proposed the low-rank matrix decomposition model and its derivation algorithm of a kind of empty spectrum constraint of band, effectively extraction time domain variation characteristic, the noise of removal diversified forms, the falseness for strengthen real change, suppressing caused by noise changes, so as to improve the change accuracy of detection of time-varying high-spectrum remote sensing.

Description

High spectrum image time varying characteristic extracting method based on low-rank decomposition and empty spectrum constraint
Technical field
The present invention relates to a kind of high spectrum image time varying characteristic extracting method based on low-rank decomposition and empty spectrum constraint.
Background technology
Remote sensing technology is the emerging complex art to grow up in the sixties in this century, with space, electron-optical, calculating The science and technology such as machine, geography are closely related, and are the powerful technique means for studying earth resource environment.High-spectrum remote-sensing be by into The multidimensional information acquiring technology being combined as technology with spectral technique.Hyperspectral imager is tens of to hundreds of electromagnetic spectrum The two-dimensional geometry space of target and one-dimensional spectral information are detected on narrow and continuous wave band simultaneously, be terrestrial object information extraction and Analysis provides extremely abundant data, so as to be widely used in geological sciences, hydrological science, precision agriculture and military neck Domain etc..
In recent years, high spectrum resolution remote sensing technique be used to carry out long-time, multi-period observation to areal, produce therewith Time-varying high spectrum image.Change detection is to identify the technology changed in multiple image, is widely used in gray level image, cromogram Picture, the processing of multi-spectral remote sensing image, and can be used for time-varying high spectrum image.Existing time-varying high spectrum image change detection Method is mainly (such as multispectral by the processing method (such as target acquisition, pixel solution mixes) of non-time-varying high spectrum image or other images Image) change detecting method improve.These methods are no lack of excellent effect person, the research to the change detection of time-varying EO-1 hyperion It is made that major contribution, such as S.Liu, L.Bruzzone, F.Bovolo, and P.Du, " Hierarchical Unsupervised change detection in multitemporal hyperspectral images, " IEEE Trans.Geosci.Remote Sens., vol.53, no.1, pp.244-260, Jan.2015. are proposed " change end member " Concept, identify that a variety of primary and secondarys change by the picture dot solution mixing method of stratification.But existing achievement in research still suffers from necessarily not Foot：First, the noise of generally existing and diversification of forms in time-varying high spectrum image is ignored, including registration error, spectrum are inclined Abnormity point and thermal noise caused by the factors such as difference, hardware limitation, so as to can not effectively suppress produced by noise " falseness change "； Second, only considered the spectral information of high spectrum image, its space characteristics or empty spectral property are not made full use of.
The content of the invention
The low-rank matrix decomposition model and its Optimization Solution algorithm constrained is composed it is an object of the invention to provide a kind of band sky, from And the time domain variation characteristic of time-varying high spectrum image is efficiently extracted, suppress abnormity point and thermal noise, improve change accuracy of detection.
In order to achieve the above object, the technical scheme is that providing a kind of based on low-rank decomposition and empty spectrum constraint High spectrum image time varying characteristic extracting method, it is characterised in that comprise the following steps：
Step 1, calculate the two groups of high-spectrum remote sensing T gathered in the same place of different time1∈RM×BAnd T2∈RM×B Differential image Y ∈ RM×B, wherein Y=T1-T2；M is sample number, i.e. number of pixels, make I and J represent respectively image line number and Columns, then there is M=I × J；B is sample dimension, i.e. wave band number；
Step 2, establish the low-rank matrix decomposition model with empty spectrum constraint：
Y=L+S+N (1)
In formula (1), L ∈ RM×BFor low-rank data, and the empty spectral property portrayed with empty spectrum constraint, for representing time domain Variation characteristic (hereinafter referred to as " time varying characteristic ")；S∈RM×BFor sparse matrix, for describing abnormity point, that is, sparse, intensity is distributed Higher noise；N∈RM×BFor Gaussian noise, the relatively low thermal noise of Gaussian distributed, intensity is typicallyed represent.Introduce variable X ∈ RM×B, obtain the Optimization Solution function of formula (1) model：
In formula (2), τ is constraint factor；ε is threshold value；|·||SSFor empty spectrum bound term proposed by the present invention：
In formula (3), xij∈RB×1For Θ X ∈ RI×J×BIn currently processed pixel, Θ is to rearrange J × B matrixes Into the operator of the rank tensors of I × J × B tri-；For with xijCentered on W × W spatial neighborhoods in remove xijOutside pixel；W WithBe constant set in advance, regulation W andValue can control the smoothness of time varying characteristic, prevented smooth；Order W is positive odd number,Space coordinates in Θ X is (in, jn), i, in∈ { 1,2 ..., I }, j, jn∈ { 1,2 ..., J }, and (in, jn) ≠ (i, j), then have
Step 3, the mode of alternating iteration is taken to solve, comprise the following steps the L of formula (1) and formula (2), S, X, N：
Step 3.1, by augmented vector approach formula (2) is converted into：
In formula (4), Λ1And Λ2It is Lagrange multiplier；μ is penalty factor.Obtained by formula (4)：
In formula (5), (6), (7), t represents iterations, and t=0,1 ..., tmax, tmaxFor the iterations upper limit；
Step 3.2, arrangement formula (5), (6), (7), (1) obtain L, S, X, N solution successively.
Preferably, the step 3.2 includes：
Step 3.2.1, convolution (5) and bilateral sciagraphy, are obtained：
In formula (8)：Wherein： A1 ∈RB×rAnd A2∈RM×rFor the bilateral projection matrix generated at random；Q1∈RM×MAnd R1∈RM×rIt is Z1QR split-matrixes, Q2∈RB ×BAnd R2∈RB×rIt is Z2QR split-matrixes；Q is the accelerated factor of bilateral projection, and r is the restriction threshold value of L order, and q and r are It is constant set in advance；
Step 3.2.2, method is limited by formula (6) and soft-threshold, obtained：
In formula (9),Operator is limited for soft-threshold, and
Step 3.2.3, for formula (7), orderX=[x1, x2..., xm..., xM]T(wherein m=1,2 ..., M, m=(j-1) × I+i, i=1,2 ..., I, j=1,2 ..., J), and make J (X) local derviation to X is zero, obtains X(t+1)In each vectorClosed solutions it is as follows：
In formula (10),For complete 1 vector；Represent X(t)In WithCentered on W × W spatial neighborhoods in pixel；ForAnd xijLocus it is identical,WithLocus it is identical, therefore the definition of spatial neighborhood in formula (10), W and ωwDefinition and w withSpace coordinates the same formula (3) of corresponding relation；
Step 3.2.4, obtained by formula (1) and formula (8) to formula (10)：
N(t+1)=Y-L(t+1)-S(t+1) (11)
Step 3.2.5, according to formula (8) to formula (11), L, X, S, N, its optimal solution of iterative are alternately updated；When iteration is missed Poor ErroriReach given threshold εi, εi＞ 0, i=1,2, or iterations t reaches tmaxWhen, iteration ends, gained L or X are Time domain variation characteristic.
Preferably, in step 3.2.5, iteration error ErroriIt is defined as follows：
Error2=| | L(t+1)-X(t+1)||∞ (13)
To sum up, the algorithm that the present invention is realized is designated as LRSD_SS, it is contemplated that Lagrange multiplier and formula in formula (4) (2) the iteration renewal of each coefficient in formula (13), the factor is arrived, it is comprised the following steps that：
A) T is inputted1∈RM×BAnd T2∈RM×B, τ, λ, r, q, ε1, ε2, tmax, μ initial value μ0, μ renewal step-length ρ, and μ threshold value μmax；
B) Y=T is calculated1-T2, and initialize：Make iterations t=0, L(0)=Y, X(0)=0, S(0)=0, Λ1 (0)=0, Λ2 (0)=0, μ=μ0；
C) L is calculated respectively by formula (8), (9), (10), (11)(t+1), X(t+1), S(t+1), N(t+1)；
D) make
E) μ ← min (ρ μ, μ are mademax)；
F) Error is calculated respectively by formula (12), (13)1, Error2；
G) t ← t+1 is made；
If h) Error1＞ ε 1, Error2＞ ε2, or t ＜ tmax, then return c)；Otherwise go to i)；
I) L=L is made(t+1), S=S(t+1), N=N(t+1), X=X(t+1), and export.
The present invention makes full use of the difference of the time varying characteristic of high spectrum image and noise in inherent data structure with timely Become the empty spectral property of part of feature, it is proposed that the low-rank matrix decomposition model and its Optimization Solution algorithm of a kind of empty spectrum constraint of band, So as to efficiently extract time domain variation characteristic, suppress abnormity point and thermal noise, improve change accuracy of detection.In the empty spectrum constraint of design When item and derivation algorithm, present invention additionally contemplates that the problems such as computation complexity and convergence, to improve algorithm LRSD_SS practicality Property.
The present invention has following features：
1) present invention makes full use of the inherent data structure characteristics of time-varying high spectrum image to design the time-varying shaped like formula (2) Feature representation model：Respectively in the low-rank component in low-rank decomposition model, sparse component, Gaussian component and differential image when Become feature, abnormity point, thermal noise and establish contact, and the part of time varying characteristic is utilized by the empty spectrum constraint shaped like formula (3) Empty spectral property, synchronously realize time varying characteristic extraction and robustness noise reduction.
2) the empty spectrum constraint reaction is the empty spectrum distance of neighborhood from rather than the space length of certain wave band pixel or certain two Spectrum intervals between pixel, therefore can effectively portray the empty spectral property of part of time varying characteristic；
3) the empty spectrum bound term includes neighborhood size W and weightsThe two parameters, can by be pre-adjusted W and Value control the local smoothing method degree of time varying characteristic, prevented smooth.
4) the empty spectrum bound term using l2 norms come portray the empty spectrum distance of neighborhood from so that X has closed solutions；The constraint Item can also synchronously utilize the spatially and spectrally information of high spectrum image, rather than carry out space smoothing by wave band, save the present invention The run time of proposed algorithm；
5) bilateral sciagraphy is employed when solving low-rank component L, its computation complexity is r2(M+3B+4r)+(4q+4) MBr；
6) X closed solutions and method of Lagrange multipliers and the good convergence of bilateral sciagraphy itself, this hair are had benefited from Bright derivation algorithm LRSD_SS possesses good convergence.
By adopting the above-described technical solution, the present invention is compared to the prior art, have the following advantages that and good effect：
1) the inherent data structure for time-varying high spectrum image carries out analysis and modeling, and designs time varying characteristic with this Extracting method：Because high spectrum image is high dimensional data (a), and earth's surface change categorical measure is limited in shooting area, Mei Zhongbian Pixel corresponding to change typically has correlation in stronger class, therefore time varying characteristic is generally present in some low-dimensional data space, With low-rank；(b) noise of time-varying high spectrum image is caused by multiple factors such as registration error, spectrum deviation, hardware limitations, Be broadly divided into abnormity point and thermal noise, the former has, and openness and intensity is higher, the latter typically follow Gaussian Profile and intensity compared with Low, noise can behave as " falseness change ", disturb the detection of real change, so establishing the expression model described in formula (2), use The low-rank matrix decomposition method that low-rank data, sparse data and thermal noise can be separated simultaneously carries out feature extraction and robustness drop Make an uproar；
2) present invention composes the immanent structure characteristic of the further mining data of bound term, such as certain by the sky described in formula (3) A little special noises have an inherent data structure (such as necrosis point has low-rank and openness concurrently) similar to time varying characteristic, and when Become the local smoothing method that then there is feature general noise not possess, so as to lift time varying characteristic extraction effect；Especially, the sky is composed Bound term reaction be neighborhood empty spectrum distance from, rather than spectrum between the space length of certain wave band pixel or certain two pixel away from From, therefore can effectively portray the empty spectral property of part of time varying characteristic；Sky spectrum constraint also allows with default and adjustment parameter Mode prevents the excessively smooth of time varying characteristic；
3) present invention considers not only the validity of time varying characteristic extraction, has also taken into account the practicality of derivation algorithm：(a) it is empty Spectrum bound term passes through l2Norm come portray neighborhood sky spectrum distance from so that X has closed solutions, and being capable of synchronization process space and light Spectrum information, rather than space smoothing is carried out by wave band, ensure the efficiency of algorithm；(b) multiplied using the good Lagrange of convergence Sub- method and bilateral sciagraphy and the empty spectrum constraint with closed solutions, so that derivation algorithm has good convergence.
Brief description of the drawings
Fig. 1 is the flow chart of the present invention；
Fig. 2 (a) is that the time domain change atural object of emulation time-varying high spectrum image is truly schemed, and wherein white represents region of variation, Black represents non-changing region；
Fig. 2 (b) and Fig. 2 (c) is respectively to emulate T in time-varying high spectrum image1∈RM×BAnd T2∈RM×BPseudocolour picture, its R, G, channel B are respectively the 10th of corresponding high spectrum image the, 20,40 wave band；
Fig. 3 is the box traction substation of the amplitude Distribution value of time varying characteristic, wherein the upper, middle and lower edge of each " case " corresponds to respectively Third quartile, second quartile (median), first quartile, from " palpus " table of upper and lower edge extension Show appropriate outlier；
Fig. 4 (a) and Fig. 4 (b) is the derivation algorithm LRSD_SS of present invention convergence curve；
Fig. 5 (a) is that the time domain change atural object of true time-varying high spectrum image is truly schemed, and wherein white represents region of variation, Black represents non-changing region；
Fig. 5 (b) and Fig. 5 (c) is respectively T in true time-varying high spectrum image1And T2Pseudocolour picture, its R, G, channel B point Not Wei corresponding high spectrum image the 20th, 40,60 wave band；
Embodiment
With reference to specific embodiment, the present invention is expanded on further.It should be understood that these embodiments are merely to illustrate the present invention Rather than limitation the scope of the present invention, in addition, it is to be understood that after the content that the present invention lectures is read, those skilled in the art Various changes and modification can be made to the present invention, these equivalent form of values equally fall within what the application appended claims were limited Scope.
Embodiments of the present invention are related to a kind of high spectrum image time varying characteristic based on low-rank decomposition and empty spectrum constraint and carried Method is taken, the algorithm comprises the following steps：Calculate the difference in the time-varying high-spectrum remote sensing of the same place collection of different time Different image；The low-rank matrix decomposition model with empty spectrum constraint is established according to the inherent data structure characteristics of differential image；Pass through friendship For each component of the iterative model, and then extract time domain variation characteristic.By gained time varying characteristic L amplitude vector A ∈ RM×1 (wherein A (m)=| | L (m,：)||F, m=1,2 ..., M) substitute into K-means or other can realize it is unsupervised two classification point In class device, A (m) can be labeled as " change class " or " non-changing class ", so as to mark off region of variation on image space domain With non-changing region, the change detection of time-varying high spectrum image is realized.
Emulate data experiment
Time-varying high spectrum image is emulated to be constructed by real non-time-varying high spectrum image " Pavia universities data ".Should Image is by http：//www.ehu.es/ccwintco/uploads/e/ee/PaviaU.mat is provided, and its acquisition time is 2001 Year, place is the Pavia universities of Italy, imager ROSIS, has 610 × 340 pixels (i.e. I=610, J=340), After removing noise pollution, water absorption bands, remaining 103 wave bands (i.e. B=103).Emulate the construction side of time-varying high spectrum image Method is as follows：1) using former Pavia universities data as the high spectrum image T sampled at the moment 110∈RM×B(M=I × J= 207400)；2) in T106 regions of middle selection, the pixel in region two-by-two is exchanged, the time domain change of atural object is simulated, when obtaining Carve the high spectrum image T of 2 samplings20∈RM×BAnd the atural object of region of variation is truly schemed (shown in such as Fig. 2 (a))；3) respectively to T10 And T20Abnormity point as described in Table 1 and thermal noise are added, obtains emulating time-varying high spectrum image T1∈RM×BAnd T2∈RM×B, it is pseudo- Cromogram is respectively as shown in Fig. 2 (b) and Fig. 2 (c).
Table 1 emulates thermal noise and abnormity point (being normalized random noise)
Grader after being extracted using K-means as time varying characteristic, realize change detection.Inventive algorithm LRSD_SS Parameter setting it is as follows：τ=0.01, μmax=106, ρ=1.05,Q=3, W=3,ε1=ε2=10-6, tmax=30, wherein i=1,2 ... I, j=1, 2 ... J.The hardware environment of all algorithm operations is Intel (R) Xeon (R) X5667CPU 3.00GHz (double-core) 24GB internal memories, Software platform is Windows 7 and MATLAB R2013b.
Test the checking of 1 algorithm effect
The evaluation of algorithm effect mainly includes following three kinds of modes：(a) the range value distribution character of time varying characteristic is investigated； (b) match stop mark figure is truly schemed with atural object；(c) overall classification accuracy (Overall Accuracy, OA), average mark are calculated Class precision (Average Accuracy, AA) and Kappa coefficients (κ).
Respectively to the change class (Change) and non-changing class (Non-Change) shown in Fig. 2 (a), investigate without disparity map of making an uproar As Y0(Y0=T10-T20), differential image Y (Y=T1-T2), time varying characteristic L obtained by inventive algorithm LRSD_SS, by W.He, H.Zhang, L.Zhang, and H.Shen, " Total-variation-regularized low-rank matrix Factorization for hyperspectral image restoration, " IEEE Trans.Geosci.Remote Sens., the low-rank matrix for the full variation space constraint of band that vol.54, no.1, pp.176-188, Jan.2016. are proposed, which is decomposed, to be calculated L obtained by method LRSD_TV and by T.Zhou and D.Tao, " GoDec：randomized low-rank&sparse matrix Decomposition in noisy case, " Proc.28th ICML, Jan.2011, pp.33-40. are proposed classical low L range value distribution character obtained by order matrix decomposition algorithm LRSD, as shown in figure 3, LRSD_SS of the present invention time varying characteristic amplitude With more close distribution within class and more scattered distribution between class, is easy to be classified device and correctly identifies.As shown in table 2, when When grader is taken as the Unsupervised clustering algorithm K-means of classics, LRSD_SS algorithms proposed by the present invention can obtain higher Nicety of grading, its change Detection results be better than in table classical (SAM and PCA) or advanced time varying characteristic extraction algorithm (ASCD, LRSD and LRSD_TV).
Time varying characteristic extraction algorithm+the K-means of table 2 change testing result (emulation time-varying high spectrum image)
It is realTest the checking of 2 efficiency of algorithm
LRSD_SS and LRSD_TV is made to solve low-rank component using bilateral sciagraphy.As shown in table 3, L and S is being solved On, the low-rank decomposition algorithm spent time of both belt restrainings is close；And on X is solved, because what LRSD_SS of the present invention was used Sky spectrum constraint causes X to have closed solutions, so (its space constraint is needed by wave band meter far fewer than LRSD_TV for the algorithm time-consuming Calculate, and without closed solutions, therefore time-consuming more).It follows that the computation complexity of the present invention is not high, run time can receive, its Efficiency is higher than advanced low-rank matrix decomposition algorithm LRSD_TV.
The single iteration of table 3 takes (unit：Second；Experimental data：Emulate time-varying high spectrum image)
Test the checking of 3 Algorithm Convergences
As shown in Fig. 4 (a) and Fig. 4 (b), algorithm LRSD_SS proposed by the present invention can receive by the iteration of 20 times or so Hold back, there is certain practical value.
True Data is tested
True time-varying high spectrum image is by c.Wu, B.Du, and L.Zhang, " A subspace-based used in this experiment Change detection method for hyperspectral images, " IEEE J.Sel.Topics.Appl.Earth Observ.Remote Sens., vol.6, no.2, pp.815-830, Apr.2013. are carried For its imager is Hyperion, T1∈RM×BAnd T2∈RM×B(wherein M=I × J=63000, I=450, J=140, B= 155) acquisition time is respectively on May 3rd, 2006 and on April 23rd, 2007, and collecting location is Jiangsu Province, China Yancheng agricultural Land used.Truly such as Fig. 5 (a) is shown for the atural object of region of variation, T1And T2Pseudocolour picture respectively as shown in Fig. 5 (b) and Fig. 5 (c). Except experimental data, this is tested remaining and set togetherEmulate data experiment。
As shown in table 4, for the true time-varying high spectrum image shown in Fig. 5, when grader is taken as the unsupervised poly- of classics During class algorithm K-means, LRSD_SS algorithms proposed by the present invention can obtain higher nicety of grading, and it changes Detection results Better than (SAM and PCA) or advanced time varying characteristic extraction algorithm (ASCD, LRSD and LRSD_TV) classical in table.
Time varying characteristic extraction algorithm+the K-means of table 4 change testing result (true time-varying high spectrum image)

Claims (5)

  1. A kind of 1. high spectrum image time varying characteristic extracting method based on low-rank decomposition and empty spectrum constraint, it is characterised in that including Following steps：
    Step 1, calculate the two groups of high-spectrum remote sensing T gathered in the same place of different time1∈RM×BAnd T2∈RM×BDifference Different image Y ∈ RM×B, wherein Y=T1-T2；M is sample number, makes I and J represent the line number and columns of image respectively, then have M=I × J；B is sample dimension；
    Step 2, establish the low-rank matrix decomposition model with empty spectrum constraint：
    Y=L+S+N (1)
    In formula (1), L ∈ RM×BFor low-rank data, and the empty spectral property portrayed with empty spectrum constraint, for representing that time domain changes Feature；S∈RM×BFor sparse matrix, for describing abnormity point；N∈RM×BFor Gaussian noise.Introduce variable X ∈ RM×B, obtain formula (1) the Optimization Solution function of the model：
    <mrow> <mtable> <mtr> <mtd> <msub> <mi>min</mi> <mrow> <mi>L</mi> <mo>,</mo> <mi>S</mi> </mrow> </msub> </mtd> <mtd> <mrow> <mo>|</mo> <mo>|</mo> <mi>L</mi> <mo>|</mo> <msub> <mo>|</mo> <mo>*</mo> </msub> <mo>+</mo> <mi>&amp;lambda;</mi> <mo>|</mo> <mo>|</mo> <mi>S</mi> <mo>|</mo> <msub> <mo>|</mo> <mn>1</mn> </msub> <mo>+</mo> <mi>&amp;tau;</mi> <mo>|</mo> <mo>|</mo> <mi>X</mi> <mo>|</mo> <msub> <mo>|</mo> <mrow> <mi>S</mi> <mi>S</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>s</mi> <mo>.</mo> <mi>t</mi> <mo>.</mo> </mrow> </mtd> <mtd> <mrow> <mo>|</mo> <mo>|</mo> <mi>Y</mi> <mo>-</mo> <mi>L</mi> <mo>-</mo> <mi>S</mi> <mo>|</mo> <msubsup> <mo>|</mo> <mi>F</mi> <mn>2</mn> </msubsup> <mo>&amp;le;</mo> <mi>&amp;epsiv;</mi> <mo>,</mo> <mi>L</mi> <mo>=</mo> <mi>X</mi> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>
    In formula (2), τ is constraint factor；ε is threshold value；|·||SSBound term is composed for sky：
    <mrow> <mo>|</mo> <mi>X</mi> <mo>|</mo> <msub> <mo>|</mo> <mrow> <mi>S</mi> <mi>S</mi> </mrow> </msub> <mo>=</mo> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>I</mi> </munderover> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>J</mi> </munderover> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>w</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <msup> <mi>W</mi> <mn>2</mn> </msup> <mo>-</mo> <mn>1</mn> </mrow> </munderover> <msubsup> <mi>&amp;omega;</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> <mi>w</mi> </msubsup> <mo>|</mo> <mo>|</mo> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>-</mo> <msubsup> <mi>x</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> <mi>w</mi> </msubsup> <mo>|</mo> <msubsup> <mo>|</mo> <mn>2</mn> <mn>2</mn> </msubsup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
    In formula (3), xij∈RB×1For Θ X ∈ RI×J×BIn currently processed pixel, Θ be by J × B matrixes be rearranged into I × The operator of the rank tensors of J × B tri-；For with xijCentered on W × W spatial neighborhoods in remove xijOutside pixel；W and Be constant set in advance, regulation W andValue can control the smoothness of time varying characteristic, prevented smooth；The W is made to be Positive odd number,Space coordinates in Θ X is (in, jn), i, in∈ { 1,2 ..., I }, j, jn∈ { 1,2 ..., J }, and (in, jn) ≠ (i, j), then have
    Step 3, the mode of alternating iteration is taken to solve, comprise the following steps the L of formula (1) and formula (2), S, X, N：
    Step 3.1, by augmented vector approach formula (2) is converted into：
    <mrow> <mtable> <mtr> <mtd> <mrow> <mi>min</mi> <mi> </mi> <mi>l</mi> <mrow> <mo>(</mo> <mi>L</mi> <mo>,</mo> <mi>S</mi> <mo>,</mo> <mi>X</mi> <mo>,</mo> <msub> <mi>&amp;Lambda;</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>&amp;Lambda;</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>=</mo> <munder> <mrow> <mi>arg</mi> <mi>min</mi> </mrow> <mrow> <mi>L</mi> <mo>,</mo> <mi>S</mi> <mo>,</mo> <mi>X</mi> <mo>,</mo> <msub> <mi>&amp;Lambda;</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>&amp;Lambda;</mi> <mn>2</mn> </msub> </mrow> </munder> <mo>|</mo> <mo>|</mo> <mi>L</mi> <mo>|</mo> <msub> <mo>|</mo> <mo>*</mo> </msub> <mo>+</mo> <mi>&amp;lambda;</mi> <mo>|</mo> <mo>|</mo> <mi>S</mi> <mo>|</mo> <msub> <mo>|</mo> <mn>1</mn> </msub> <mo>+</mo> <mi>&amp;tau;</mi> <mo>|</mo> <mo>|</mo> <mi>X</mi> <mo>|</mo> <msub> <mo>|</mo> <mrow> <mi>S</mi> <mi>S</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mo>&lt;</mo> <msub> <mi>&amp;Lambda;</mi> <mn>1</mn> </msub> <mo>,</mo> <mi>Y</mi> <mo>-</mo> <mi>L</mi> <mo>-</mo> <mi>S</mi> <mo>&gt;</mo> <mo>+</mo> <mo>&lt;</mo> <msub> <mi>&amp;Lambda;</mi> <mn>2</mn> </msub> <mo>,</mo> <mi>X</mi> <mo>-</mo> <mi>L</mi> <mo>&gt;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mfrac> <mi>&amp;mu;</mi> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <mo>|</mo> <mo>|</mo> <mi>Y</mi> <mo>-</mo> <mi>L</mi> <mo>-</mo> <mi>S</mi> <mo>|</mo> <msubsup> <mo>|</mo> <mi>F</mi> <mn>2</mn> </msubsup> <mo>+</mo> <mo>|</mo> <mo>|</mo> <mi>X</mi> <mo>-</mo> <mi>L</mi> <mo>|</mo> <msubsup> <mo>|</mo> <mi>F</mi> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>
    In formula (4), Λ1And Λ2It is Lagrange multiplier；μ is penalty factor, is obtained by formula (4)：
    <mrow> <mtable> <mtr> <mtd> <mrow> <msup> <mi>L</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </msup> <mo>=</mo> <munder> <mrow> <mi>arg</mi> <mi>min</mi> </mrow> <mi>L</mi> </munder> <mo>|</mo> <mo>|</mo> <mi>L</mi> <mo>|</mo> <msub> <mo>|</mo> <mo>*</mo> </msub> <mo>+</mo> <mo>&lt;</mo> <msubsup> <mi>&amp;Lambda;</mi> <mn>1</mn> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </msubsup> <mo>,</mo> <mi>Y</mi> <mo>-</mo> <mi>L</mi> <mo>-</mo> <msup> <mi>S</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </msup> <mo>&gt;</mo> <mo>+</mo> <mo>&lt;</mo> <msubsup> <mi>&amp;Lambda;</mi> <mn>2</mn> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </msubsup> <mo>,</mo> <msup> <mi>X</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </msup> <mo>-</mo> <mi>L</mi> <mo>&gt;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mfrac> <mi>&amp;mu;</mi> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <mo>|</mo> <mo>|</mo> <mi>Y</mi> <mo>-</mo> <mi>L</mi> <mo>-</mo> <msup> <mi>S</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </msup> <mo>|</mo> <msubsup> <mo>|</mo> <mi>F</mi> <mn>2</mn> </msubsup> <mo>+</mo> <mo>|</mo> <mo>|</mo> <msup> <mi>X</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </msup> <mo>-</mo> <mi>L</mi> <mo>|</mo> <msubsup> <mo>|</mo> <mi>F</mi> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <munder> <mrow> <mi>arg</mi> <mi>min</mi> </mrow> <mi>L</mi> </munder> <mo>|</mo> <mo>|</mo> <mi>L</mi> <mo>|</mo> <msub> <mo>|</mo> <mo>*</mo> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mi>&amp;mu;</mi> <mo>|</mo> <mo>|</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <mi>Y</mi> <mo>+</mo> <msup> <mi>X</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </msup> <mo>-</mo> <msup> <mi>S</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </msup> <mo>+</mo> <mfrac> <mn>1</mn> <mi>&amp;mu;</mi> </mfrac> <mo>(</mo> <mrow> <msubsup> <mi>&amp;Lambda;</mi> <mn>1</mn> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>&amp;Lambda;</mi> <mn>2</mn> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </msubsup> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>-</mo> <mi>L</mi> <mo>|</mo> <msubsup> <mo>|</mo> <mi>F</mi> <mn>2</mn> </msubsup> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <msup> <mi>S</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </msup> <mo>=</mo> <munder> <mrow> <mi>arg</mi> <mi>min</mi> </mrow> <mi>S</mi> </munder> <mi>&amp;lambda;</mi> <mo>|</mo> <mo>|</mo> <mi>S</mi> <mo>|</mo> <msub> <mo>|</mo> <mn>1</mn> </msub> <mo>+</mo> <mfrac> <mi>&amp;mu;</mi> <mn>2</mn> </mfrac> <mo>|</mo> <mo>|</mo> <mi>S</mi> <mo>-</mo> <mrow> <mo>(</mo> <mi>Y</mi> <mo>-</mo> <msup> <mi>L</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </msup> <mo>+</mo> <mfrac> <msubsup> <mi>&amp;Lambda;</mi> <mn>1</mn> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </msubsup> <mi>&amp;mu;</mi> </mfrac> <mo>)</mo> </mrow> <mo>|</mo> <msubsup> <mo>|</mo> <mn>2</mn> <mn>2</mn> </msubsup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <msup> <mi>X</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </msup> <mo>=</mo> <munder> <mrow> <mi>arg</mi> <mi>min</mi> </mrow> <mi>X</mi> </munder> <mi>J</mi> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> <mo>=</mo> <munder> <mrow> <mi>arg</mi> <mi>min</mi> </mrow> <mi>X</mi> </munder> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>|</mo> <mo>|</mo> <mi>X</mi> <mo>-</mo> <mi>Q</mi> <mo>|</mo> <msubsup> <mo>|</mo> <mn>2</mn> <mn>2</mn> </msubsup> <mo>+</mo> <mfrac> <mi>&amp;tau;</mi> <mi>&amp;mu;</mi> </mfrac> <mo>|</mo> <mo>|</mo> <mi>X</mi> <mo>|</mo> <msub> <mo>|</mo> <mrow> <mi>S</mi> <mi>S</mi> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>
    In formula (5), (6), (7), t represents iterations, and t=0,1 ..., tmax, tmaxFor the iterations upper limit；
    Step 3.2, arrangement formula (5), (6), (7), (1) obtain L, S, X, N solution successively.
  2. A kind of 2. high spectrum image time varying characteristic extraction side based on low-rank decomposition and empty spectrum constraint as claimed in claim 1 Method, it is characterised in that the step 3.2 includes：
    Step 3.2.1, convolution (5) and bilateral sciagraphy, are obtained：
    <mrow> <msup> <mi>L</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </msup> <mo>=</mo> <msub> <mi>Q</mi> <mn>1</mn> </msub> <msup> <mrow> <mo>&amp;lsqb;</mo> <msub> <mi>R</mi> <mn>1</mn> </msub> <msup> <mrow> <mo>(</mo> <msubsup> <mi>A</mi> <mn>2</mn> <mi>T</mi> </msubsup> <msub> <mi>Z</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msubsup> <mi>R</mi> <mn>2</mn> <mi>T</mi> </msubsup> <mo>&amp;rsqb;</mo> </mrow> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>q</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </msup> <msubsup> <mi>Q</mi> <mn>2</mn> <mi>T</mi> </msubsup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>
    In formula (8)：Wherein： A1 ∈RB×rAnd A2∈RM×rFor the bilateral projection matrix generated at random；Q1∈RM×MAnd R1∈RM×rIt is Z1QR split-matrixes, Q2∈RB ×BAnd R2∈RB×rIt is Z2QR split-matrixes；Q is the accelerated factor of bilateral projection, and r is the restriction threshold value of L order, and q and r are It is constant set in advance；
    Step 3.2.2, method is limited by formula (6) and soft-threshold, obtained：
    In formula (9),Operator is limited for soft-threshold, and
    Step 3.2.3, for formula (7), orderX=[x1, x2..., xm..., xM]T(wherein m=1,2 ..., M, m=(j-1) × I+i, i=1,2 ..., I, j=1,2 ..., J), and J (X) is made to X Local derviation be zero, obtain X(t+1)In each vectorClosed solutions it is as follows：
    <mrow> <msubsup> <mi>x</mi> <mi>m</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <mfrac> <mrow> <msup> <mover> <mi>Q</mi> <mo>~</mo> </mover> <mrow> <mi>m</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </msup> <mi>v</mi> </mrow> <mrow> <msup> <mi>v</mi> <mi>T</mi> </msup> <mn>1</mn> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow>
    In formula (10),For complete 1 vector； Represent X(t)In withFor Pixel in W × W spatial neighborhoods at center；For And xij's Locus is identical,WithLocus it is identical, therefore in formula (10) spatial neighborhood definition, W and ωwDefinition and w WithSpace coordinates the same formula (3) of corresponding relation；
    Step 3.2.4, obtained by formula (1) and formula (8) to formula (10)：
    N(t+1)=Y-L(t+1)-S(t+1) (11)
    Step 3.2.5, according to formula (8) to formula (11), L, X, S, N, its optimal solution of iterative are alternately updated；Work as iteration error ErroriReach given threshold εi, εi＞ 0, i=1,2, or iterations t reaches tmaxWhen, iteration ends, when gained L or X are Domain variation characteristic.
  3. A kind of 3. high spectrum image time varying characteristic extraction side based on low-rank decomposition and empty spectrum constraint as claimed in claim 2 Method, it is characterised in that in step 3.2.5, iteration error ErroriIt is defined as follows：
    <mrow> <msub> <mi>Error</mi> <mn>1</mn> </msub> <mo>=</mo> <mfrac> <mrow> <mo>|</mo> <mo>|</mo> <mi>Y</mi> <mo>-</mo> <msup> <mi>L</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </msup> <mo>-</mo> <msup> <mi>S</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </msup> <mo>|</mo> <msub> <mo>|</mo> <mi>F</mi> </msub> </mrow> <mrow> <mo>|</mo> <mo>|</mo> <mi>Y</mi> <mo>|</mo> <msub> <mo>|</mo> <mi>F</mi> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow>
    Error2=| | L(t+1)-X(t+1)||∞ (13)。
  4. A kind of 4. high spectrum image time varying characteristic extraction side based on low-rank decomposition and empty spectrum constraint as claimed in claim 1 Method, it is characterised in that by be pre-adjusted and set W andValue can control the local smoothing method degree of time domain variation characteristic.
  5. A kind of 5. high spectrum image time varying characteristic extraction side based on low-rank decomposition and empty spectrum constraint as claimed in claim 1 Method, it is characterised in that the algorithm LRSD_SS realized includes Lagrange multiplier in formula (4), and formula (2) is arrived in formula (13) The iteration renewal of all coefficients, the factor, is comprised the following steps that：
    A) T is inputted1∈RM×BAnd T2∈RM×B, τ, λ, r, q, ε1, ε2, tmax, μ initial value μ0, μ renewal step-length ρ, and μ Threshold value μmax；
    B) Y=T is calculated1-T2, and initialize：Make iterations t=0, L(0)=Y, X(0)=0, S(0)=0, Λ1 (0)=0, Λ2 (0) =0, μ=μ0；
    C) L is calculated respectively by formula (8), (9), (10), (11)(t+1), X(t+1), S(t+1), N(t+1)；
    D) make
    E) μ ← min (ρ μ, μ are mademax)；
    F) Error is calculated respectively by formula (12), (13)1, Error2；
    G) t ← t+1 is made；
    If h) Error1＞ ε1, Error2＞ ε2, or t ＜ tmax, then return c)；Otherwise go to i)；
    I) L=L is made(t+1), S=S(t+1), N=N(t+1), X=X(t+1), and export.
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